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Curl of a Vector Field
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Gradient of a Scalar Field
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Cartesian Coordinate
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Definition of a Matrix
Current Location
>
Math Formulas
>
Linear Algebra
> Scalar Multiplication
Scalar Multiplication
The matrix obtained by multiplying every element of a matrix A by a scalar λ is called the scalar multiple of A by λ.
For example:
If
Then the product of 3A will be:
Properties of Scalar Multiplication:
All the laws of ordinary algebra hold for the addition or subtraction of matrices and their multiplication by scalars.
· If
A
and
B
be two matrices of the same order and if
k
be a scalar, then:
k (A + B) = kA + kB
· If
k
_{1}
and
k
_{2}
are two scalars and if
A
is a matrix, then:
(k
_{1}
+ k
_{2}
)A = k
_{1}
A + k
_{2}
A and k
_{1}
(k
_{2}
A) = k
_{2}
(k
_{1}
A)
Example 1:
Let
, find the product of 4
A
.
Example 2:
Find out whether or not
c
(
Ax
) =
A
(
cx
) is a valid equation, where
c
is a scalar,
A
is a 2 by 2 matrix, and
x
is a dimension 2 column vector?
Answer:
Yes,
c
(
Ax
) =
A
(
cx
) is a valid equation, because the order of multiplication does not matter for real numbers.
If
,
and the scalar
c
= c, we have:
and
We can see that
c(Ax)
and
A(cx)
both produces the same matrix at the end. Hence
c
(
Ax
) =
A
(
cx
).
Note:
c
(
Ax
) =
A
(
cx
) holds true for all matrices that have the correct dimensions so that the product of
Ax
is valid.
Example 3:
Find the product of
.
Solution:
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